Tomorrow's my day off training and there's a B&Q just behind my workplace, so I'll be heading over there to see what I can acquire.

Any advice appreciated, will update thread if I manage to construct something workable.

I've tried pretty much what you describe, but with different cores (two types of pvc pip, cheap gardening bamboo). There's a trade off between stiffness/durability and protecting your partner. You can only do so much with padding until they become unwieldy.

core:
If you use plastic pipe, experiment with different stiffnesses (water outlet, oil pipe, etc.).
I tried a pack of bamboo stakes from the gardening section of the diy store, but they were way too thin and flimsy. If you can find some thick rattan, great.
I'm going to try dowels next for the core, which is what we found in some of the ones we bought and broke on each other.

padding:
Foam insulation is fine in my experience, but you want to be careful with what you do with the tip. Cut the core a little shorter than the length you want so you can add a foam tip for starters.

cladding:
Duck tape or cheaper tape has worked ok for me. Some people don't like the little edges you can get at the end. You can finish it off with some electrical tape, or a taped on sock.

water proofing:
One brand of stick that we bought and broke on each other had a plastic membrane between the foam and fabric padding. I think it was to protect the foam. I wouldn't bother with that.

handle:
Some brands have grips like bike handles. I haven't figured out if the complication is worth it myself.

We use PVC pipe wrapped with insulation. The duck tape around them. We make a handle by leaving that part without any insulation and then wrapping that part with grip tape. They last forever, if the insulation breaks down, just tape over.

They are heavier than regular sticks but that just makes for a good workout.

Tried wooden dowels. They require too much padding to stop being dangerous at any decent power. By the time they are soft, they are too thick. Make them a comfortable thickness, and you can hardly tell they are padded.

One varriation was thick soft rubber tubing (dryer hose) around a wooden dowel. This produced a fantastically brutal fighting stick, but was terrible as a "safety" device.

Rattan, Bamboo or some type of plastic piping all make for decent cores. Liugtweight and durable. Both bamboo and pipe can produce some very UNSAFE byproducts when banged around so keep this in mind when wrapping them. You want any chips or sharp edges thoroughly contained.

A cheap source of foam is those "pool noodles". (you can get these at the dollar store) They even have a hollow in the middle for your stick. They are thick as all get out, but compress fairly easily when taping. A little elbow grease and you can get them to a managable girth. The sheer amount of condensed padding when doing this ensures they remain pretty darn soft even though most of the air has been squeezed out. The pool noodle and rattan are probably your best bets for weight, strength and safety.

If you use PVC or the like, try to find a product like nylon. IE, something that doesn't shatter easily. I've wound up with a few "floppy" stick failures due to brittle core materail.

Also, making the padding extende beyond the tip of the core is VERY important. This is one of the hardest parts to get right. I don't know why, it seems so simple, but I swear it's a total pain making these safe enough for hard jabs and punyo shots. I like to leave 1"-2" inches or so, and fill in the remaining space with scrap foam or even cork from a bottle.

If you cant find pool noodles, or you find they are to thick for your liking, you can get tube shaped foam insulation for hoses and piping at the hardware store. This stuff is only @ 1/2" thick though, so you may need to layer.

Mine is pretty similar, 3/8 cpvc (external diameter of 1/2" or so) or 1/2 nylon rod for light sticks, 3/4" nylon rod for heavy padded sticks. I like the heavier far better, they hit with far more authority and get rid of the whippy little hits. I use pipe insulation for the padding. For the end I use a simple little closed cell foam endcap to prevent the other person from being hit by the core. Looks like this:

Oh, and to stop that annoying sound of the core bouncing around inside the insulation, I tape the core to the padding using little tubes of duct tape flattened to the core, you could also use double sided tape.

I've used PVC, tape and pipe insulation to make a desquerdes a few years ago:
and it held up to stickwork pretty well actually. On the other hand, when it was starting to fall apart and I was done with it, I was able to break it into manageable pieces easily with some hardwood power strikes.

I have a pair of old padded sticks from PI in the late 90s. WHen they were new they were thin rattan wrapped in pipe insulation inside a nylon cover. The cores have broken several times over the years and I have tried everything mentioned here except the nylon rods. The last repair job I used PVC then I filled the PVS with silicone caulk and dried it in the sun a few days. Two years later and they are still holding up, they are safer than rattan but you can put a pretty good whip on them. I have had complaints from those hit with them, but no injuries other than a bruise.

So one of the guys at our gym's made us some really awesome sticks with wooden dowels inside some sort of really light metal sheathing inside pipe insulation wrapped in duct tape. (They look pretty much exactly like blindside's.

They hurt like a mofo, but are still soft enough that they make for "safe" training. (To be honest, considering my goal of eventually fighting at a Dog Brothers gathering, I prefer the strikes I take to be painful enough that I learn to avoid them).